Properties

Degree 2
Conductor $ 2^{3} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.88·3-s − 1.71·5-s − 4.47·7-s + 5.35·9-s − 4.06·11-s − 4.73·13-s + 4.96·15-s − 17-s − 8.07·19-s + 12.9·21-s + 4.47·23-s − 2.05·25-s − 6.79·27-s − 1.34·29-s + 5.63·31-s + 11.7·33-s + 7.68·35-s + 9.06·37-s + 13.6·39-s + 8.49·41-s + 0.753·43-s − 9.19·45-s − 4.24·47-s + 13.0·49-s + 2.88·51-s − 1.55·53-s + 6.98·55-s + ⋯
L(s)  = 1  − 1.66·3-s − 0.767·5-s − 1.69·7-s + 1.78·9-s − 1.22·11-s − 1.31·13-s + 1.28·15-s − 0.242·17-s − 1.85·19-s + 2.82·21-s + 0.934·23-s − 0.410·25-s − 1.30·27-s − 0.250·29-s + 1.01·31-s + 2.04·33-s + 1.29·35-s + 1.49·37-s + 2.19·39-s + 1.32·41-s + 0.114·43-s − 1.37·45-s − 0.619·47-s + 1.86·49-s + 0.404·51-s − 0.213·53-s + 0.941·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8024,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 + T \)
good3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + 1.71T + 5T^{2} \)
7 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
19 \( 1 + 8.07T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 1.34T + 29T^{2} \)
31 \( 1 - 5.63T + 31T^{2} \)
37 \( 1 - 9.06T + 37T^{2} \)
41 \( 1 - 8.49T + 41T^{2} \)
43 \( 1 - 0.753T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + 1.55T + 53T^{2} \)
61 \( 1 + 8.96T + 61T^{2} \)
67 \( 1 - 2.40T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 6.99T + 73T^{2} \)
79 \( 1 + 0.499T + 79T^{2} \)
83 \( 1 + 0.659T + 83T^{2} \)
89 \( 1 + 4.70T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.37799215141721542571037379202, −6.56728455679561933243288013064, −6.22033867618674314845209632335, −5.49860430646856564235487885963, −4.60176613805979276168621908334, −4.27332520067681738231289442801, −3.06347340453918951905178581853, −2.35630154641327929987662769472, −0.57394683759065298038374972875, 0, 0.57394683759065298038374972875, 2.35630154641327929987662769472, 3.06347340453918951905178581853, 4.27332520067681738231289442801, 4.60176613805979276168621908334, 5.49860430646856564235487885963, 6.22033867618674314845209632335, 6.56728455679561933243288013064, 7.37799215141721542571037379202

Graph of the $Z$-function along the critical line